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Dirac mass induced by optical gain and loss

Abstract

Mass is commonly considered an intrinsic property of matter, but modern physics reveals particle masses to have complex origins1, such as the Higgs mechanism in high-energy physics2,3. In crystal lattices such as graphene, relativistic Dirac particles can exist as low-energy quasiparticles4 with masses imparted by lattice symmetry-breaking perturbations5,6,7,8. These mass-generating mechanisms all assume Hermiticity, or the conservation of energy in detail. Using a photonic synthetic lattice, we show experimentally that Dirac masses can be generated by means of non-Hermitian perturbations based on optical gain and loss. We then explore how the spacetime engineering of the gain and loss-induced Dirac mass affects the quasiparticles. As we show, the quasiparticles undergo Klein tunnelling at spatial boundaries, but a local breaking of a non-Hermitian symmetry can produce a new flux non-conservation effect at the domain walls. At a temporal boundary that abruptly flips the sign of the Dirac mass, we observe a variant of the time-reflection phenomenon: in the non-relativistic limit, the Dirac quasiparticle reverses its velocity, whereas in the relativistic limit, the original velocity is retained.

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Fig. 1: Scheme for realizing a non-Hermitian synthetic lattice.
Fig. 2: Evolution of a Gaussian wave packet in the synthetic lattice.
Fig. 3: Klein tunnelling of massive Dirac quasiparticles.
Fig. 4: Time reflection and refraction of Dirac quasiparticles.

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Data availability

The experimental data are available in the data repository for Nanyang Technological University at https://doi.org/10.21979/N9/0KBPTS. Other data supporting the findings of this study are available from the corresponding authors on reasonable request.

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Acknowledgements

This work was supported by the Singapore Ministry of Education (MOE) Tier 1 Grant No. RG148/20, Singapore Ministry of Education Academic Research Fund Tier 2 Grant No. MOE-T2EP50123-0007 and by the National Research Foundation (NRF), Singapore under Competitive Research Programme NRF-CRP23-2019-0007 and NRF-CRP23-2019-0005, and NRF Investigatorship NRF-NRFI08-2022-0001. C.S. and R.G. acknowledge the support of the Quantum Engineering Programme of the Singapore National Research Foundation, grant number NRF2021-QEP2-01-P01. H.X. acknowledges the support of the start-up fund and the direct grant (Grant No. 4053675) from The Chinese University of Hong Kong.

Author information

Authors and Affiliations

Authors

Contributions

H.X., B.Z. and Y.D.C. conceived the idea. H.X., L.Y., R.G. and E.A.C. designed and performed the experiment. L.Y. and R.G. analysed the data. C.S., B.Z. and Y.D.C. supervised the project. H.X., L.Y., R.G., Y.Y.T., C.S., B.Z. and Y.D.C. contributed to the discussion of the results and writing of the manuscript.

Corresponding authors

Correspondence to Cesare Soci, Baile Zhang or Y. D. Chong.

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Extended data figures and tables

Extended Data Fig. 1 Schematic of the experiment.

The abbreviations are as follows: BPF, band-pass filter; CW, continuous wave; EDFA, erbium-doped fibre amplifier; EOM, electro-optical modulator; PBS, polarization beam splitter; PD, photodetector; PM, phase modulator. The boxes containing arrows denote isolators and the ovals labelled ‘50/50’ and ‘90/10’ denote optical couplers with the indicated splitting ratios.

Extended Data Fig. 2 Floquet band diagrams of the synthetic lattice for different gain/loss and phase distributions.

a, No gain/loss (g = 0). b, Gain/loss/gain/loss. The Dirac point turns into a pair of EPs. The band energies are non-real at k = 0, ϕ = π, whereas the band energies close to k = ±π, ϕ = π are real. c, Gain/loss/loss/gain. At k = 0, ϕ = π, there are two orthogonal bands with real band energies, governed by a massive Dirac Hamiltonian. In b and c, the gain/loss level is g = 0.41.

Extended Data Fig. 3 Dispersion relation for the Dirac quasiparticles.

These Floquet band diagrams are calculated using the evolution equations (orange) and the effective Dirac Hamiltonian (blue; see Supplementary Information), for g = 0 (left) and g = 0.41 (right). a, Quasienergies versus k. b, Quasienergies versus ϕ.

Extended Data Fig. 4 Preparation of pulse train for exciting quasiparticles.

a, Time evolution of a single initial pulse under equations (20)–(23), for ϕ0 = −π. b, Intensity profile at m = 55.

Extended Data Fig. 5 Optical energy evolution and Fourier spectra for different gain/loss and phase distributions.

a, No gain/loss (g = 0), ϕ = π. b, Gain/loss/gain/loss, ϕ = π, g = 0.4. c, Gain/loss/loss/gain at k = 0, ϕ = π, g = 0.4. In ac, the simulated intensity evolutions are obtained with a single pulse injection in the long loop at time m = 0. In each subplot, the Fourier spectra are derived from Fourier transformation of the simulation results on the left. All the other parameters are the same as these in experiments.

Extended Data Fig. 6 Pulse propagation in the symmetry-broken regime.

a, Raw data showing the evolution of a wave packet centred at k = π, with g = 0.4055, ϕ = π and uniform loss rate γ = 0.223. b, Plot of the total intensity in the short loop versus m.

Extended Data Fig. 7 Boundary states induced by gain and loss.

a, Schematic of a synthetic lattice with two domains separated by a boundary (dashes). The effective Dirac Hamiltonians have mass M < 0 and M > 0 in the left and right domains, respectively. Here the colour represents the pseudospin of the Dirac quasiparticle, which is derived from the inner product between the eigenvector and the eigenvector of a massless Dirac Hamiltonian (the branch with positive group velocity). b, Floquet band diagram for a finite sample of the lattice shown in a, with total size N = 59 and gain/loss level g = 0.59. A chiral dispersion relation, corresponding to gain/loss-induced boundary states, spans the gap. c, Spatial distribution of |un|2 + |vn|2 for the mid-gap boundary state, calculated with different values of g. d, Spatial profiles of \(| {u}_{n}^{m}{| }^{2}+| {v}_{n}^{m}{| }^{2}\) for the theoretical boundary eigenstate for g = 0.41 (red line) and the corresponding experimental data time-averaged over 40 ≤ m ≤ 70 (blue dots). In c and d, we normalize the maximum intensity for individual subplots to 1. eg, Time evolution of a pulse injected at n = 0, for g = 0.22, 0.41 and 0.59. In simulations (left panels) and experimental data (right panels), the pulse spreads into a wavefunction localized at the boundary, with localization length following the trend shown in c. In cg, we set ϕ = π.

Extended Data Fig. 8 Pulse propagation at different potential barrier heights.

a, Schematic of the lattice configuration near the interface. b, Transmission and reflection coefficient as a function of potential V. ce, Simulated intensity evolution as the potential barrier varies from 0 to 0.2π. In ce, we set ϕ = π, g = 0.4 (which corresponds to M ≈ 0.08) and k = 0.1π (which corresponds to E ≈ 0.03π).

Extended Data Fig. 9 Pulse profile at time step m = 42.

Total intensity distribution with Gaussian pulse excitations of different momentum k. A temporal boundary is introduced at m = 10 for the right subplot. For all figures, we set ϕ = π and g = 0.48.

Supplementary information

Supplementary Information

This file contains Supplementary Notes 1–4, including Supplementary Figs. 1–5 and further references.

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Yu, L., Xue, H., Guo, R. et al. Dirac mass induced by optical gain and loss. Nature (2024). https://doi.org/10.1038/s41586-024-07664-x

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